Some inequalities of matrix power and Karcher means for positive linear maps

TL;DR

This paper extends matrix inequalities involving matrix power and Karcher means for positive definite matrices, providing bounds under certain conditions with applications to positive linear maps.

Contribution

It generalizes existing inequalities for matrix power and Karcher means, introducing new bounds involving positive unital linear maps and parameters.

Findings

Derived new inequalities for matrix power and Karcher means.

Established bounds involving positive unital linear maps.

Provided conditions under which inequalities hold.

Abstract

In this paper, we generalize some matrix inequalities involving matrix power and Karcher means of positive definite matrices. Among other inequalities, it is shown that if ${\mathbb A}=(A_{1},...,A_{n})$ is a $n$-tuple of positive definite matrices such that $0<m\leq A_{i}\leq M\, (i=1,\cdots,n)$ for some scalars $m< M$ and $\omega=(w_{1},\cdots,w_{n})$ is a weight vector with $w_{i}\geq0$ and $\sum_{i=1}^{n}w_{i}=1$, then \begin{align*} \Phi^{p}\Big(\sum_{i=1}^{n}w_{i}A_{i}\Big)\leq \alpha^{p}\Phi^{p}(P_{t}(\omega; {\mathbb A})) \end{align*} and \begin{align*} \Phi^{p}\Big(\sum_{i=1}^{n}w_{i}A_{i}\Big)\leq \alpha^{p}\Phi^{p}(\Lambda(\omega; {\mathbb A})), \end{align*} where $p>0$, $\alpha=\max\Big\{\frac{(M+m)^{2}}{4Mm}, \frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big\}$, $\Phi$ is a positive unital linear map and $t\in [-1, 1]\backslash \{0\}$.

Equations

$$\displaystyle\Phi^{p}\Big{(}\sum_{i=1}^{n}w_{i}A_{i}\Big{)}\leq\alpha^{p}\Phi^{p}(P_{t}(\omega;{\mathbb{A}}))$$

$$\displaystyle\Phi^{p}\Big{(}\sum_{i=1}^{n}w_{i}A_{i}\Big{)}\leq\alpha^{p}\Phi^{p}(\Lambda(\omega;{\mathbb{A}})),$$

$$\frac{A+B}{2}\ge A♯B.$$

$$X=\sum _{i=1}^n{w}_i(X{♯}_t{A}_i),t\in (0,1]$$

$${\left(\sum _{i=1}^n{w}_i{A}_i^{-1}\right)}^{-1}\le P_t(\omega ;\mathbb{A})\le \sum _{i=1}^n{w}_i{A}_i$$

$$\sum _{i=1}^n{w}_i\log \left(X^{\frac{1}{2}}{A}_i^{-1}{X}^{\frac{1}{2}}\right)=0.$$

$$\underset{t\to 0}{\lim }{P}_t(\omega ;\mathbb{A})=\Lambda (\omega ;\mathbb{A})$$

$${\left(\sum _{i=1}^n{w}_i{A}_i^{-1}\right)}^{-1}\le \Lambda (\omega ;\mathbb{A})\le \sum _{i=1}^n{w}_i{A}_i.$$

$$A^p\ge B^p(0\le p\le 1).$$

$$\Phi (A{)}^{-1}\le \Phi (A^{-1}).$$

$$\Phi (A♯B)\le \Phi (A)♯\Phi (B).$$

$$\Phi (A)♯\Phi (B)\le \frac{M+m}{2\sqrt{Mm}}\Phi (A♯B).$$

$$\displaystyle(\Phi(A)\sharp\Phi(B))^{p}\leq\Big{(}\frac{M+m}{2\sqrt{Mm}}\Big{)}^{p}\Phi^{p}(A\sharp B),\,\,\,(0<p\leq 1).$$

$$\Phi (A^{-1})\le \frac{(M+m{)}^2}{4Mm}\Phi (A{)}^{-1}$$

$$\Phi (P_t(\omega ;\mathbb{A}))\le P_t(\omega ;\Phi (\mathbb{A})),$$

$$\displaystyle P_{t}^{2}(\omega;\Phi({\mathbb{A}}))\leq\Big{(}\frac{(m+M)^{2}}{4mM}\Big{)}^{2}\Phi^{2}(P_{t}(\omega;{\mathbb{A}}))$$

$$\displaystyle P_{t}^{p}(\omega;\Phi({\mathbb{A}}))\leq\Big{(}\frac{(m+M)^{2}}{4mM}\Big{)}^{p}\Phi^{p}(P_{t}(\omega;{\mathbb{A}})),\,\,\,(0<p\leq 2)$$

$$\displaystyle P_{t}^{p}(\omega;\Phi({\mathbb{A}}))\leq\Big{(}\frac{(m+M)^{2}}{4^{\frac{2}{p}}mM}\Big{)}^{p}\Phi^{p}(P_{t}(\omega;{\mathbb{A}}))$$

$$\displaystyle\Big{\|}P_{t}^{\frac{p}{2}}(\omega;\Phi({\mathbb{A}}))\Phi^{-\frac{p}{2}}(P_{t}(\omega;{\mathbb{A}}))\Big{\|}\leq\frac{(m+M)^{p}}{4M^{\frac{p}{2}}m^{\frac{p}{2}}}.$$

$$\displaystyle M^{\frac{p}{2}}m^{\frac{p}{2}}\Big{\|}P_{t}^{\frac{p}{2}}(\omega;\Phi({\mathbb{A}}))\Phi^{-\frac{p}{2}}(P_{t}(\omega;{\mathbb{A}}))\Big{\|}$$

$$\displaystyle\leq\frac{1}{4}\Big{\|}P_{t}^{\frac{p}{2}}(\omega;\Phi({\mathbb{A}}))+M^{\frac{p}{2}}m^{\frac{p}{2}}\Phi^{-\frac{p}{2}}(P_{t}(\omega;{\mathbb{A}}))\Big{\|}^{2}$$

$$(\text{by Lemma ref6(i)})$$

$$\displaystyle\leq\frac{1}{4}\Big{\|}(P_{t}(\omega;\Phi({\mathbb{A}}))+Mm\Phi^{-1}(P_{t}(\omega;{\mathbb{A}})))^{\frac{p}{2}}\Big{\|}^{2}$$

$$(\text{by Lemma ref6(ii)})$$

$$=\frac{1}{4}\Vert (P_t(\omega ;\Phi (\mathbb{A}))+Mm{\Phi }^{-1}(P_t(\omega ;\mathbb{A}))){\Vert }^p$$

$$\le \frac{1}{4}{\Vert \sum _{i=1}^n{w}_i\Phi (A_i)+Mm\Phi (P_t(\omega ;\mathbb{A}{)}^{-1})\Vert }^p$$

$$(\text{by eqref10})$$

$$\le \frac{1}{4}{\Vert \sum _{i=1}^n{w}_i\Phi (A_i)+Mm\Phi (\sum _{i=1}^n{w}_i{A}_i^{-1})\Vert }^p$$

$$(\text{by eqref11})$$

$$\displaystyle=\frac{1}{4}\left\|\sum_{i=1}^{n}w_{i}\Big{(}\Phi(A_{i})+Mm\Phi(A_{i}^{-1})\Big{)}\right\|^{p}.$$

$$Mm\Phi (A_i^{-1})+\Phi (A_i)\le M+m(i=1,2,\cdots ,n)$$

$$||P_t^{\frac{p}{2}}(\omega ;\Phi (\mathbb{A})){\Phi }^{-\frac{p}{2}}(P_t(\omega ;\mathbb{A}))||\le \frac{(m+M{)}^p}{4M^{\frac{p}{2}}{m}^{\frac{p}{2}}}.$$

$$\displaystyle(\Phi(A)\sharp_{\alpha}\Phi(B))^{p}\leq\Big{(}\frac{(m+M)^{2}}{4^{\frac{2}{p}}mM}\Big{)}^{p}\Phi^{p}(A\sharp_{\alpha}B),$$

$$\displaystyle\Lambda^{p}(\omega;\Phi({\mathbb{A}}))\leq\Big{(}\frac{(m+M)^{2}}{4^{\frac{2}{p}}mM}\Big{)}^{p}\Phi^{p}(\Lambda(\omega;{\mathbb{A}}))$$

$$\sum _{i=1}^n{w}_i{A}_i\le \frac{(M+m{)}^2}{4Mm}{P}_t(\omega ;\mathbb{A}),$$

Full text

[TABLE]

and

[TABLE]

where $p>0$, \alpha=\max\Big{\{}\frac{(M+m)^{2}}{4Mm},\frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big{\}}, $\Phi$ is a positive unital linear map and $t\in[-1,1]\backslash\{0\}$.

1. Introduction and preliminaries

Let $\mathcal{M}_{n}$ be the $C^{*}$-algebra of all $n\times n$ complex matrices and $\langle\,.\,,\,.\,\rangle$ be the standard scalar product in $\mathbb{C}^{n}$ with the identity $I$. For Hermitian matrices $A,B\in\mathcal{M}_{n}$, we write $A\geq 0$ if $A$ is positive semidefinite, $A>0$ if $A$ is positive definite, and $A\geq B$ if $A-B\geq 0$. If $m,M$ be real scalars, then we mean $m\leq A\leq M$ that $mI\leq A\leq MI$.

The Gelfand map $f(t)\mapsto f(A)$ is an isometrical $*$-isomorphism between the $C^{*}$-algebra $C({\rm sp}(A))$ of continuous functions on the spectrum ${\rm sp}(A)$ of a Hermitian matrix $A$ and the $C^{*}$-algebra generated by $A$ and $I$. If $f,g\in C({\rm sp}(A))$, then $f(t)\geq g(t)\,\,(t\in{\rm sp}(A))$ implies that $f(A)\geq g(A)$. A linear map $\Phi$ on $\mathcal{M}_{n}$ is positive if $\Phi(A)\geq 0$ whenever $A\geq 0$. It is said to be unital if $\Phi(I)=I$. A norm $|||\cdot|||$ on $\mathcal{M}_{n}$ is said to be unitarily invariant norm if $|||UAV|||=|||A|||$, for all unitary matrices $U$ and $V$.

Let $A,B\in\mathcal{M}_{n}$ be two positive definite and $t\in[0,1]$. The operator t-weighted arithmetic, geometric, and harmonic means of $A,B$ are defined by $A\nabla_{t}B=(1-t)A+tB$, $A\sharp_{t}B=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{t}A^{\frac{1}{2}}$ and $A!_{t}B=((1-t)A^{-1}+tB^{-1})^{-1}$ respectively, in which $A!_{t}B\leq A\sharp_{t}B\leq A\nabla_{t}B.$ In particular, for $t=\frac{1}{2}$ we get the operator arithmetic mean $\nabla$, the geometric mean $\sharp$ and the harmonic mean $!$. The AM-GM inequality reads

[TABLE]

In [12], Lim and Palfia have introduced matrix power means of positive definite matrices of some fixed dimension. If ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ is a $n$-tuple of positive definite matrices $A_{i}\,\,(i=1,\cdots,n)$ and $\omega=(w_{1},\cdots,w_{n})$ is a positive probability weight vector where $w_{i}\geq 0\,\,(i=1,\cdots,n)$ and $\sum_{i=1}^{n}w_{i}=1$, then the matrix power means $P_{t}(\omega;{\mathbb{A}})$ is defined to be the unique positive definite solution of the non-linear equation:

[TABLE]

For $t\in[-1,0)$, it is defined by $P_{t}(\omega;{\mathbb{A}})=P_{-t}(\omega;{\mathbb{A}^{-1}})^{-1}$, where ${\mathbb{A}}^{-1}=(A_{1}^{-1},\cdots,A_{n}^{-1})$.

We denote $P_{1}(\omega;{\mathbb{A}})=\sum_{i=1}^{n}w_{i}A_{i}$ and $P_{-1}(\omega;{\mathbb{A}})=(\sum_{i=1}^{n}w_{i}A_{i}^{-1})^{-1}$, the weighted arithmetic and harmonic means of $A_{1},\cdots,A_{n}$, respectively.

There is one of important properties of matrix power means $P_{t}(\omega;{\mathbb{A}})$, that $P_{t}(\omega;{\mathbb{A}})$ interpolates between the weight harmonic and arithmetic means:

[TABLE]

for all $t\in[-1,1]\backslash\{0\}$.

The Karcher means of $n$ positive probability vectors in ${\mathbb{R}^{n}}$ convexity spanned by the unit coordinate vectors, is defined as the unique positive definite solution of the equation:

[TABLE]

The Karcher means denoted by $\Lambda(\omega;{\mathbb{A}})$, where it follows from (1.3) that $\Lambda(\omega;{\mathbb{A}}^{-1})^{-1}=\Lambda(\omega;{\mathbb{A}})$. It is well known that (see [12])

[TABLE]

and

[TABLE]

For further information about the matrix power mean, Karcher mean and their properties, we refer the readers to [12, 11, 13] and references therein.

It is well known that for the two positive definite matrices $A,B$, if $A\geq B$, then

[TABLE]

In general (1.6) is not true for $p>1$. Let $\Phi$ be a unital positive linear map. The following inequality is known as the Choi inequality see [5, 9].

[TABLE]

Ando [1] proved that if $\Phi$ is a positive linear map, then for positive definite matrices $A,B\in\mathfrak{B}(\mathcal{H})$ we have

[TABLE]

A reverse of the Ando’s inequality (1.8) is as follows: If $A,B\in\mathcal{M}_{n}$ and $0<m\leq A,B\leq M$, Then

[TABLE]

By inequality (1.6) we get

[TABLE]

Marshal and Olkin [16] proved that a counterpart of Choi’s inequality (1.7) as follows

[TABLE]

for positive definite $A$ with $0<m\leq A\leq M$. In addition Lin [14] and Fu [7] improved inequality (1.10) for $p\geq 2$.

The matrix power means satisfy the following inequality: for each $t\in(0,1]$

[TABLE]

where ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices and $\Phi({\mathbb{A}})=(\Phi(A_{1}),\\ \cdots,\Phi(A_{n}))$.

Dehghani et al. [6] established counterparts of (1.11) involving matrix power means as following:

[TABLE]

for all $t\in[-1,1]\backslash\{0\}$ and $0<m\leq A_{i}\leq M$.

Using inequality (1.6) we get

[TABLE]

It is interesting to ask whenever the inequality (1.12) is true for $p\geq 2$. This is the first motivation of this paper. moreover, we improve inequality (1.9) for $p\geq 2$. We also obtain some reverses of (1.2). In the last section, we establish several refinements of obtained inequalities.

2. Main results

To prove our first result, we need the following lemmas.

Lemma 2.1**.**

[4, 2, 3, 8*]** Let $A,B\in\mathcal{M}_{n}$ be positive definite matrices and $\alpha>0$. Then

$\rm{(i)}\,\,||AB||\leq\frac{1}{4}||A+B||^{2}.$

$\rm{(ii)}\,\,||A^{\alpha}+B^{\alpha}||\leq||(A+B)^{\alpha}||.$

$\rm{(iii)}\,\,A\leq\alpha B$ if and only if $||A^{\frac{1}{2}}B^{-\frac{1}{2}}||\leq\alpha^{\frac{1}{2}}.$

$\rm{(iv)}\,$ If $0\leq A\leq B$ and $0<m\leq A\leq M$, then $A^{2}\leq\frac{(M+m)^{2}}{4Mm}B^{2}.$*

Lemma 2.2**.**

[10]** Let $A\in\mathcal{M}_{n}$ be positive definite. Then $A\leq tI$ if and only if $\|A\|\leq t$ if and only if \left[\begin{array}[]{cc}tI&A\\ A^{*}&tI\end{array}\right] is positive.

Theorem 2.3**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m<M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector. If $\Phi$ is a unital positive linear map, then

[TABLE]

for every $p\geq 2$ and $t\in[-1,1]\backslash\{0\}$.

Proof.

By Lemma 2.1(iii), inequality (2.1) is equivalent to

[TABLE]

Hence, it is enough to prove inequality (2.2). So

[TABLE]

It follows from $0<m\leq A_{i}\leq M$ that $(M-A_{i})(m-A_{i})A_{i}^{-1}\leq 0\,\,(i=1,2,\cdots,n)$. Hence

[TABLE]

Using inequalities (2.3) and (2.4) we get

[TABLE]

Thus, this completes the proof. ∎

In the following result we state that inequality (1.9) is valid for any $p\geq 2$.

Corollary 2.4**.**

Let $A,B\in\mathcal{M}(\mathbb{C})$ be positive definite matrices such that $0<m\leq A,B\leq M$ for some scalars $m<M$ and $\alpha\in[0,1]$. Then

[TABLE]

for any $p\geq 2$ and unital positive linear map $\Phi$.

Proof.

Using this fact $P_{t}(1-\alpha,\alpha;A,B)=A\sharp_{\alpha}B,\ (\alpha\in[0,1])$ and $n=2,w_{1}=1-\alpha$ and $w_{2}=\alpha$ in inequality(2.1), we get the desired result. ∎

Corollary 2.5**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a n-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m<M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector. If $\Phi$ is a unital positive linear map, then

[TABLE]

for every $p\geq 2$ and $t\in[-1,1]\backslash\{0\}$.

Proof.

The proof follows from Theorem 2.3 and relation (1.4). ∎

Theorem 2.6**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a n-tuple of positive definite matrices such that $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m<M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector. Then

[TABLE]

where $t\in[-1,1]\backslash\{0\}$.

Proof.

If we put $\Phi(A)=\sum_{i=1}^{n}w_{i}A_{i}$, then for $t\in(0,1]$ we have

[TABLE]

Therefore

[TABLE]

Inequality (2.5) follows from a similar fashion for $t\in[-1,0)$. ∎

Remark 2.7*.*

As special case for $\mathbb{A}=(A,B)$ and $\omega=(w_{1},w_{2})$ with $w_{1}=w_{2}=\frac{1}{2}$, we have the following inequality:

[TABLE]

which is counterpart of AM-GM inequality (1.1).

Corollary 2.8**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a n-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m\leq M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector. Then

[TABLE]

where $t\in[-1,1]\backslash\{0\}$.

Remark 2.9*.*

Inequalities (2.5) and (2.6) can be regarded as a counterpart of inequalities (1.2) and (1.5), respectively. By inequalities (2.5) and (1.11), we can obtain the following operator inequality

[TABLE]

Now, by applying inequality (1.6) we get

[TABLE]

for $0<p\leq 1$.

In the next theorem, we show that inequality (2.8) is valid for $p>1$.

Theorem 2.10**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m<M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector. Then

[TABLE]

*where $t\in[-1,1]\backslash\{0\}$, $p>1$ and \alpha=\max\Big{\{}\frac{(M+m)^{2}}{4Mm},\frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big{\}}. *

Proof.

First we show inequality (2.9) for $p=2$. We have

[TABLE]

whence

[TABLE]

Hence

[TABLE]

Therefore

[TABLE]

Now, we prove inequality (2.9) for $p>2$. In this case we have

[TABLE]

Hence

[TABLE]

Thus

[TABLE]

Now, if we take \alpha=\max\Big{\{}\frac{(M+m)^{2}}{4Mm},\frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big{\}}, then by (2.10) and (2.11) we get the desired result. ∎

Remark 2.11*.*

By letting ${\mathbb{A}}=(A,B)$ and $\omega=(w_{1},w_{2})$ with $w_{1}=w_{2}=\frac{1}{2}$ in Theorem 2.10, the following inequalities are hold:

[TABLE]

Which appeared in [9, Theorem 4]. where \alpha=\max\Big{\{}\frac{(M+m)^{2}}{4Mm},\frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big{\}}.

Corollary 2.12**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m\leq M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector, and let $t\in[-1,1]\backslash\{0\}$. Then

[TABLE]

where $p\geq 1$ and \alpha=\max\Big{\{}\frac{(M+m)^{2}}{4Mm},\frac{(M+m)^{2}}{4^{\frac{2}{p}}Mm}\Big{\}}.

In the next result we extend inequalities (2.1) and (2.9) to the follwing form.

Theorem 2.13**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m\leq M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector, let $t\in[-1,1]\backslash\{0\}$ and $\Phi$ be a positive unital linear map. Then

[TABLE]

and

[TABLE]

where $p>0$ and \alpha=\max\Big{\{}\frac{(m+M)^{2}}{4mM},\frac{(m+M)^{2}}{4^{\frac{1}{p}}mM}\Big{\}}.

Proof.

By inequality (1.12) and Lemma 2.1(iii) for $0<p\leq 1$ we have

[TABLE]

We put $\alpha=\frac{(m+M)^{2}}{4mM}$. Using Lemma 2.2 we get

[TABLE]

and

[TABLE]

are positive. Hence

[TABLE]

is positive. Using Lemma 2.2 we get

[TABLE]

For $p>1$, using inequality (2.1) with the same argument, we get the desired inequality.

Inequality (2.13) is proved by using Theorem 2.10 and a similar method. ∎

Corollary 2.14**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m\leq M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector, $\Phi$ be a positive unital linear map. Then

[TABLE]

and

[TABLE]

where $p>0$ and \alpha=\max\Big{\{}\frac{(m+M)^{2}}{4mM},\frac{(m+M)^{2}}{4^{\frac{1}{p}}mM}\Big{\}}.

In the next result, we would like to obtain unitary invariant norm inequality involving matrix power means.

Proposition 2.15**.**

*Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m\leq M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector, and let $|||\cdot|||$ be an unitary invariant norm. Then for $t\in(0,1]$

$|||P_{t}(\omega;{\mathbb{A}})|||\leq\sum_{i=1}^{n}w_{i}|||A_{i}|||$        and      |||P_{-t}(\omega;{\mathbb{A}})|||\geq\Big{(}\sum_{i=1}^{n}w_{i}|||A_{i}^{-1}|||\Big{)}^{-1}.*

Proof.

Let $X=P_{t}(\omega;{\mathbb{A}})$. Then

[TABLE]

which implies that $|||P_{t}(\omega;{\mathbb{A}})|||\leq\sum_{i=1}^{n}w_{i}|||A_{i}|||$. For second inequality, it follows from $|||A^{-1}|||\geq|||A|||^{-1}$ for any $A>0$ that

[TABLE]

∎

3. Some refinements

In this section, we give a refinement of inequality (2.9). This inequality can be refined by a similar method that known in [17].

Theorem 3.1**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m\leq M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector, and let $t\in[-1,1]\backslash\{0\}$. Then for every positive unital linear map $\Phi$

[TABLE]

where $p\geq 2$ and $K=\frac{(M+m)^{2}}{4mM}$.

Proof.

For $p\geq 2$, we have

[TABLE]

Hence

[TABLE]

Since (3.2) is equivalent to (3.1), so inequality (3.1) holds. ∎

Remark 3.2*.*

If we put ${\mathbb{A}}=(A,B)$ and $\omega=(w_{1},w_{2})$ with $w_{1}=w_{2}=\frac{1}{2}$ in Theorem 3.1, then we get [17, Theorem 2.6] as follows:

[TABLE]

Theorem 3.3**.**

Let ${\mathbb{A}}=(A_{1},\cdots,A_{n})$ be a $n$-tuple of positive definite matrices with $0<m\leq A_{i}\leq M,\ (i=1,\cdots,n)$ for some scalars $m\leq M$ and $\omega=(w_{1},\cdots,w_{n})$ a weight vector, and let $t\in[-1,1]\backslash\{0\}$. Then for every positive unital linear map $\Phi$

[TABLE]

where $p\geq 2$ and $K=\frac{(M+m)^{2}}{4mM}$.

Proof.

For $p\geq 2$, we have

[TABLE]

Therefore

[TABLE]

Since the last inequality is equivalent to (3.3), thus this complets the proof. ∎

Remark 3.4*.*

As special case for $\mathbb{A}=(A,B)$ and $\omega=(w_{1},w_{2})$ with $w_{1}=w_{2}=\frac{1}{2}$, Theorem 3.3 is a refinement of Corollary 2.4.